Graph theory: adjacency vs incident

Usually one speaks of adjacent vertices, but of incident edges.

Two vertices are called adjacent if they are connected by an edge.

Two edges are called incident, if they share a vertex.

Also, a vertex and an edge are called incident, if the vertex is one of the two vertices the edge connects.

If for two vertices $A$ and $B$ there is an edge $e$ joining them, we say that $A$ and $B$ are adjacent.

If two edges $e$ and $f$ have a common vertex $A$, the edges are called incident.

If the vertex $A$ is on edge $e$, the vertex $A$ is often said to be incident on $e$.

There is unfortunately some variation in usage. So you need to check the particular book or notes for the definition being used.

Excerpted from wikipedia:

• Two edges of a graph are called adjacent (sometimes coincident) if they share a common vertex.

• Similarly, two vertices are called adjacent if they share a common edge.

• An edge and a vertex on that edge are called incident.

This terminology seems very sensible to my ear.

An edge "e" in a graph (Undirected or directed ) that is associated with the pair of vertices n and q is said to be incident on n and q, and n and q are said to be incident on e and to be adjacent vertices.